-32-
z'ero, and when this is the case the numerical coefficient of (13)
becomes the same as that of (18); so that all the terms of (18) are
included in (13), except for sign., Therefore we can obtain hyju..r
by selecting all the terms of f44..r in which each s has only cne
subscript not zero, making all the signs positive.
The k's are defipned by the equation

TITI—"T;"%T“'T = 14 hyx 4 kogy + Rox® 4 kyyxy 4 kogy® 4 v0eee
et

1
II[1 - vy(x))

By using formula (17) and noticing the negative signs of ay, by,e*ee,

This is a reciprocal of the form where y= 2%, 2= %8, etc

when r=1, k, =38,

when r=2, ko + ko = -El'»[(S,' + )+ (254,)] 3 .

when r =3, kg + ky3 + ooy = .51!.[(:," + 35155 4 355) + (651303 + 6533) + (65001)];
etc.

The definition of the k's is equivalent to saying that kjg.r is the
sug of all terms which contain as factors j of the a's, n of the

b's, ete. Then if ky,,,, is expressed in terms of the s's, the

indices of the term containing (s; )"'u--(sj ) pust

Jangeery QMgesryg
satisfy the conditions 2nyy =34, Sngmg=m, ***°, Sngry=r, the same
conditions which belong to formula (13). Now if we develop the
above reciprocal in terms of the s's by using (17) and supposing
y=x% 2=1% etc., all the terms contain products of s's of the
type just mentioned, so that every term belongs to one of the k's,
The numerical coefficient given by (17) is the same as that of (13)
except for sign. In the above reciprocal, every a, b, ¢, etc. has
with it a minus sign, so that every ("J;:"r)" in (17) must be
multiplied by (—1)“(j e il (—1)"‘"1) . When this is done
every term becomes positive, Using the above cenditions, k..

is obtained fromw # ..r by making all the sig’né positive,